對於一個n階遞迴關係式的相關問題，許多研究利用生成函數、特徵方程式的形式進行討論，本篇研究將以矩陣的形式輔以特徵方程式進行遞迴關係式的相關討論，並且利用矩陣的相關性質探討有著不同性質的遞迴關係式；本篇研究將以Perrin數列作為研究的開頭，並且定義新的數列Q_n數列，嘗試利用Q_n數列找尋Perrin數列的相關性質，接著將會推廣Perrin數列探討k階Perrin數列以及k階Q_n數列{T_n}的表示方式及特徵值的性質，藉此建立k階遞迴關係式在特定情況下的相關性質。

For solving n-th recurrence relations ,many researches use formal power series or characteristic polynomial. In this research,we use matrix methods with characteristic polynomial to discuss different properties in recurrence relations.This topic discuss Perrin sequence at first. Next,we define a new sequence “Q_n sequence”,and try to find the relation between “Perrin sequence” and “ Q_n sequence”.Finally, we extend “Perrin sequence” and “ Q_n sequence” the 3-th recurrence relations to the n-th recurrence relations,and we named them as “n-step Perrin sequence” and  “n-step  Q_n sequence,{T_n}  ”.

目錄
1. 三階遞迴關係式利用矩陣解決方法-1
2. 三階遞迴關係式Perrin數列性質探討-7
3. 三階遞迴關係式Q_n數列性質探討-11
4. Perrin數列的延伸探討-18
5. Q_n數列介紹的延伸探討-25
參考文獻-32

Contents
1.The Method of Solving the 3-Step Recurrence Relations-33
2. The Discussion of 3-Step Recurrence Relaion Perrin Sequence-39
3. The Discussion of 3-Step Recurrence Relation Q_n-Sequence-43
4. Extension Concept of Perrin Sequence-50
5. The Introduction of k-Step Q_n-Sequence-57
References-64

H. Prodinger, “Some information about the binomial transform,”The Fibonacci Quarterly, vol. 32, no. 5, pp. 412–415, 1994.